This document provides information on addition, subtraction, multiplication, and division of integers. It begins by explaining that when adding or multiplying integers with the same sign, you keep the same sign, and with different signs, the result is negative. Examples are provided to illustrate addition, subtraction, multiplication, and division of integers. The document then discusses properties of integer operations like closure, commutativity, associativity, distributivity, identity, and inverses. Activities are included for students to practice integer operations.

1. Christian Allan Alferez

2. ADDITION OF SAME SIGNED
INTEGERS
• When integers have the same
sign, simply add the integers.
• The sum will have the same
sign as the integers.

3. EXAMPLES
+ퟔ + +ퟕ
= +ퟏퟑ
+ퟏퟏ + +ퟓ
= +ퟏퟔ

4. ADDITION OF DIFFERENT
SIGNED INTEGERS
• When integers have different
sign, find the difference
between two numbers.
• The sum will have the sign of
the integer with a larger
absolute value.

5. EXAMPLES
+ퟕ + −ퟓ
= +ퟕ − ퟓ
= +ퟐ
−ퟏퟓ + +ퟕ
= −ퟏퟓ + +ퟕ
= −ퟖ

6. Christian Allan Alferez

7. ADDITION USING NUMBER
LINE
Activity: Using the number line,
find the sum of the following:
1. 5 & 4
2. 6 & 11
3. 23 & -25
4. -17 & 21
5. -13 & -3

8. ADDITION USING SIGNED TILES
• This is another device that can
be used to represent integers.
+
• The tile represents
integer 1, the tile -
represents
integer -1.

9. 1. 6+1
EXAMPLES
Solution:
+ + + + + + +
+
6(+1) + 1(+1)
=7

10. 2. (-3)+4
EXAMPLES
Solution:
- - - + + + +
+
3(-1) + 4(+1)
=1

11. Activity: Find the sum of the following signed
tiles (Column A) on its corresponding value
(Column B):
+ + +
____ 1. + a. 7
____ 2. - - + + + +
b .-4
____ 3. + c. 3
+ + + + +
+ +
- - - -
____ 4. + d. 1
____ 5. + e. 0
+ + + +
- - - -

12. SEATWORK
1. Mrs. Reyes charged P3752 worth of
groceries on her credit card. Find her
balance after she made a payment of
P2530.
2. In a game, Team Azkals lost 5 yards in
one play but gained 7 yards in the next
play. What was the actual yardage gain of
the team?

13. 3. A vendor gained P50.00 on the first day;
lost P28.00 on the second day, and
gained P49.00 on the third day. How
much profit did the vendor gain in 3 days?
4. Ronnie had PhP2280 in his checking
account at the beginning of the month. He
wrote checks for PhP450, P1200, and
PhP900. He then made a deposit of
PhP1000. If at any time during the month
the account is overdrawn, a PhP300
service charge is deducted. What was
Ronnie’s balance at the end of the
month?

14. ASSIGNMENT
Using the number line, find the
sum of the following:
1. 6 & 3
2. -40 & 11
3. 1 & --1
4. -15 & 8
5. -9 and -8

15. Using the signed tiles, find the sum
of the following:
1. 5 & 3
2. -3 & -3
3. 1 & 4
4. -1 & -6
5. --5 & -1

16. Christian Allan Alferez

17. MENTAL MATH!
Give the sum:
1. 53 + 25 6. 25 + 43
2. (-6) + 123 7. (-30) + (-20)
3. (-4) + (-9) 8. (-19) + 2
4. 6 + 15 9. 30 + (-9)
5. 16 + (-20) 10. (-19) + (-15

18. • In subtracting integers, change
the sign or find the additive
inverse of the subtrahend, then
proceed to addition.

19. EXAMPLES
+ퟔ − +ퟕ
= +ퟔ + −ퟕ
= −ퟏ
+ퟏퟏ − +ퟓ
= +ퟏퟏ + −ퟓ
= ퟔ

20. Christian Allan Alferez

21. GROUP ACTIVITY
Using the number line, find the
difference of the following:
1. 8 & 18
2. 6 & 3
3. 1 & --1
4. 16 & -7
5. -8 & -10

22. Using the signed tiles, find the
difference of the following:
1. 6 & 2
2. -3 & -3
3. 3 &1
4. -5 & 3
5. 6 & -6

23. SEATWORK
1. Maan deposited P53400.00 in her
account and withdrew P19650.00 after a
week. How much of her money was left in
the bank?
2. Two trains start at the same station at the
same time. Train A travels 92km/h, while
train B travels 82km/h. If the two trains
travel in opposite directions, how far apart
will they be after an hour? If the two trains
travel in the same direction, how far apart
will they be in two hours?

24. 3. During the Christmas season, the student
gov’t association was able to solicit 2,356
grocery items and was able to distribute
2,198 to one barangay. If this group
decided to distribute 1,201 grocery items
to the next barangay, how many more
grocery items did they need to solicit?

25. ASSIGNMENT
Read the rules in multiplying
integer and we will have a graded
recitation.

26. Christian Allan Alferez

27. MULTIPLICATION OF SAME
SIGNED INTEGERS
• When integers have the same
sign, simply multiply the
absolute value of the integers.
• The product of same signed
integers is always positive.

28. MULTIPLICATION OF
DIFFERENT SIGNED INTEGERS
• When integers have different
signs, simply multiply the
absolute value of the integers.
• The product of different signed
integers is always negative.

29. EXAMPLES
1. 3 cars with 4 passengers each,
how many passengers in all?
4 x 3 = 4 + 4 + 4 = 12

30. 2. 4 cars with 3 passengers each,
how many passengers in all?
3 x 4 = 4 x 3 =3 + 3 + 3 + 3 = 12
3. When a boy loses P6 for 3
consecutive days, what is his total
loss?
(-6) + (-6) + (-6) = (-6) (3) = -18

31. ACTIVITY
(MATH DILEMMA)
How can a person fairly divide 10
apples among 8 children so that
each child has the same share?
To solve the dilemma, match the
letter in column II with the number
that corresponds to the numbers in
column I.

32. Column I
____1. (6) (-12)
____2. (-13) (-13)
____3. (19)(-17)
____4. (-15)(29)
____5. (165)(0)
____6. (-18)(-15)
____7. (-15)(-20)
____8. (-5)(-5)(-5)
____9. (-2)(-2)(-2)(-2)
____10. (4)(6)(8)
Column II
C. 270
P. -72
E. 300
K. -323
A. -435
M. 0
L. 16
J. -125
U. 169
I. 192

33. SEATWORK
1. Jof has twenty P5 coins in her coin
purse. If her niece took 5 of the
coins, how much has been taken
away?
2. Mark can type 45 words per minute,
how many words can Mark type in
30 minutes?

34. ASSIGNMENT
What was the original name for the
butterfly?
To find the answer, find the
quotient of each of the following
and write the letter of the letter of
the problems in the box
corresponding to the quotient.

35. Y ퟏퟒퟒ ÷ (−ퟑ)
R −ퟑퟓퟐ ÷ ퟐퟐ
T ퟏퟐퟖ ÷ ퟏퟔ
E ퟏퟔퟖ ÷ ퟔ
L −ퟒퟒퟒ ÷ −ퟏퟐ
U −ퟏퟐퟎ ÷ ퟖ
T −ퟏퟒퟕ ÷ ퟕ
B ퟏퟎퟖ ÷ ퟗ
F −ퟑퟏퟓ ÷ (−ퟑퟓ)
9 37 -15 -8 -21 28 -16 12 -48

36. Christian Allan Alferez

37. DIVISION OF SAME SIGNED
INTEGERS
• When integers have the same
sign, simply divide the absolute
value of the integers.
• The quotient of same signed
integers is always positive.
• If possible, express the quotient
in lowest term.

38. EXAMPLES
+ퟔ ÷ +ퟕ
=
ퟔ
ퟕ
−ퟐ ÷ −ퟒ
=
ퟐ
ퟒ
=
ퟏ
ퟐ

39. DIVISION OF DIFFERENT
SIGNED INTEGERS
• When integers have different
signs, simply divide the absolute
value of the integers.
• The quotient of different signed
integers is always negative.
• If possible, express the quotient
in lowest term.

40. Note: However, division by zero is
not possible.

41. ACTIVITY
Perform the indicated operations
1. ퟐ − ퟑ × (−ퟒ)
2. ퟒ × ퟓ + ퟕퟐ ÷ −ퟔ
3. ퟗ + ퟔ − −ퟑ × ퟏퟐ ÷ (−ퟗ)

42. ASSIGNMENT
Review the operations of integers
and be ready for a quiz.

43. Christian Allan Alferez

44. CLOSURE PROPERTY
• When two integers is multiplied
or added, the result is also
belongs to Z.
a, b ∈ Z, then a + b ∈ Z, a∙b ∈ Z

45. EXAMPLE
Z= {…-3, -2, -1, 0, 1, 2, 3 …}
It is closed to:
• Addition
• Multiplication
• Subtraction

46. COMMUTATIVE PROPERTY
• Any order of two integers that
are either added or multiplied
does not change the value of
sum or product.
For addition
a + b = b + a
For multiplication
ab = ba

47. EXAMPLES
6+3 = 3+6
4x5 = 5x4

48. ASSOCIATIVE PROPERTY
• Any grouping of two integers
that are either added or
multiplied does not change the
value of sum or product.

49. EXAMPLES
(6+3)+5 =6+(3+5)
(3∙4)5 = 3 (4∙5)

50. DISTRIBUTIVE PROPERTY
• When two numbers have been
added or subtracted and then
multiplied by a factor, the result will
be the same when each number is
multiplied by the factor and the
products and then added or
subtracted.
a(b + c) = ab + ac

51. EXAMPLES
6 (3+5) =6(3) + 6(5)
5 (8 – 6) = 5(8) – 5(6)

52. IDENTITY PROPERTY
Additive Identity
• The sum of any number and 0 is the
given number.
• Zero (0) is the additive identity.
a + 0 = a
Multiplicative Identity
• The product of any number and 1 is the
given number.
• One (1) is the multiplicative identity.
a ∙1 = a

53. EXAMPLES
1 + 0 = 1
3∙1 = 3

54. INVERSE PROPERTY
Additive Inverse
• The sum of any number and its additive
inverse is zero.
• -a is the additive inverse of the number a.
a + (-a) = 0
Multiplicative Inverse
• The product of any number and its
multiplicative inverse is one.
•
1
a
is the multiplicative inverse of the
number a.
a ∙
1
a
= 1

55. EXAMPLES
4 + (-4) = 0
5 x
1
5
= 1

56. ACTIVITY
Complete the Table: Which
property of real number justifies
each statement?

57. Given Property
1. 0 + (-3) = -3
2. 2(3 - 5) = 2(3) - 2(5)
3. (- 6) + (-7) = (-7) + (-6)
4. 1 x (-9) = -9
5. -4 x (−
1
4
)= 1
6. 2 x (3 x 7) = (2 x 3) x 7
7. 10 + (-10) = 0
8. 2(5) = 5(2)
9. 1 x −
1
4
= −
1
4
10. (-3)(4 + 9) = (-3)(4) + (-3)(9)

58. ASSIGNMENT
Fill in the blanks and determine
what properties were used to solve
the equations.
1. 5 x ( ____ + 2) = 0
2. -4 + 4 = _____
3. -6 + 0 = _____
4. (-14 + 14) + 7 = _____
5. 7 x (_____ + 7) = 49

59. Franz Jeremiah G. Ibay

60. I = {… -3, -2, -1, 0, 1, 2, 3…}
W = {1, 2, 3, 4, 5…}
D = {0.5, 07, -0.01, 0.6666….}
F = {⅓, ⅔, ⅕, ⅙, ½, ¾}

61. RATIONAL
NUMBERS

62. EXAMPLES OF RATIONAL
NUMBERS
6 =
6
1
-3 =
9
3
0.124 =
124
1000
1
2

64. • Rational numbers can be located on
the real number line.
• A number line is a visual
representation of the numbers from
negative infinity to positive infinity,
which means it extends indefinitely
in two directions.

65. • It consists of negative numbers on
its left, zero in the middle, and
positive numbers on its right.

66. EXAMPLES OF RATIONAL
NUMBERS IN THE NUMBER
LINE
Example 1: Locate 1/4 on the number
line.
a. Since 0 < 1/4 < 1, plot 0 and 1 on
the number line.

67. b. Divide the segment into 4 equal
parts.
c. The 1st mark from 0 is the point
1/4.

68. Example 2: Locate 1.75 on the number
line.
a. The number 1.75 can be written
as 7/4, and 1 < 7/4 < 2. Divide the
segment from 0 to 2 into 8 equal
parts.

69. b. The 7th mark from 0 is the point
1.75.

70. Determine whether the following
numbers are rational numbers or not.
_____1. -3 _____4. √36
_____2. π _____5. ∛6
_____3.
3
5
_____6. 2.65

71. If the number is rational, locate them
on the real number line by plotting:

72. ASSIGNMENT
Name one rational number x that
satisfies the descriptions below:
a.
1
4
< x <
1
3
b. 3 < x < π
c. -
1
8
< x < -
1
9
d.
1
10
< x <
1
2
e. -10 < x < -9

73. Franz Jeremiah G. Ibay

74. Change the following rational numbers
in fraction form or mixed number form
to decimal form:
1. −
1
4
= _____ 4.
5
2
= _____
2.
3
10
= _____ 5. −
17
10
= _____
3. 3
5
100
= _____ 6. −2
1
5
= _____

75. Change the following rational numbers
in decimal form to fraction form.
1. 1.8 = _____ 4. -0.001 = _____
2. -3.5 = _____ 5. 10.999= _____
3. -2.2 = _____ 6. 0.11 = _____

76. DECIMAL FRACTIONS
• A decimal fraction is a fraction
whose denominator is a power
of 10.

77. EXAMPLES
1
4
=
25
100
= 0.25
1
2
-4
= -4
5
10
= -4.5

78. Consider the number
1
8
.
1000 is the smallest power of 10
that is divisible by 10.
1
8
=
125
1000
= 0.125

79. NON - DECIMAL
FRACTIONS
• A non-decimal fraction is a
fraction whose denominator is
cannot be expressed as a
power of 10, which results to a
non-terminating but repeating
decimals.

80. EXAMPLES
1.
9
11
Perform the long division.
9
11
= 0.8181818181…
= 0.81

81. EXAMPLES
2. −
1
3
Perform the long division.
-
1
3
= 0.33333…
= 0.33

82. CHANGING NON-TERMINATING
BUT
REPEATING DECIMAL FORM
Example: Change the following
into its fraction forms.
1. 0. 2
2. -1.35

83. CHANGING NON-TERMINATING
BUT
REPEATING DECIMAL FORM
Example: Change the following
into its fraction forms.
1. 0. 2
2. -1.35

84. SOLUTIONS
1. Let r = 0.2222…
10r = 2.2222…
Note: Since there is only one repeated
digit, multiply the first equation
by 10.
Subtract the first equation from the
second equation:
9r = 2.0
r =
2
9

85. 1. Let r = -1.353535…
100r = -135.353535…
Note: Since there is two repeated digit,
multiply the first equation by 100.
Subtract the first equation from the
second equation:
99r = -134
r = -
134
99
= −1
35
99

86. Franz Jeremiah G. Ibay

87. Find the sum or difference of the
following.
1.
3
5
+
1
5
= _____
2.
1
8
+
5
8
= _____
3.
10
11
−
3
11
= _____
4. 3
6
7
−1
2
7
=_____

88. TO ADD OR SUBTRACT
FRACTION WITH THE SAME
DENOMINATOR
If a, b and c ∈ Z, and b ≠ 0, then
a
b
+
c
b
=
a + c
b
and
a
b
−
c
b
=
a − c
b
If possible, reduce the answer to
lowest term.

89. TO ADD OR SUBTRACT
FRACTION WITH DIFFERENT
DENOMINATOR
With different denominators,
a
b
and
c
d
, b
≠ 0 and d ≠ 0, if the fractions to be
added or subtracted are dissimilar
• Rename the fractions to make them
similar whose denominator is the
least common multiple of b and d.

90. • Add or subtract the numerators
of the resulting fractions.
• Write the result as a fraction
whose numerator is the sum or
difference of the numerators
and whose denominator is the
least common multiple of b and
d.
• If possible, reduce the result in
lowest term.

91. EXAMPLES
Addition:
a.
3
7
+
2
7
=
3 + 2
7
=
5
7
b.
2
5
+
1
4
=
8 + 5
20
=
13
20
LCD/LCM of 5 and 4 is 20.

92. EXAMPLES
Subtraction:
a.
5
7
−
2
7
=
5 − 2
7
=
3
7
b.
4
5
−
1
4
=
16 − 5
20
=
11
20
LCD/LCM of 5 and 4 is 20.

93. Give the number asked for.
1. What is three more than three and
one-fourth?
2. Subtract from 15
1
2
the sum of
2
2
3
and 4
2
5
. What is the result?
3. Increase the sum of 6
3
14
and 2
2
7
by
3
1
2
. What is the result?

94. ASSIGNMENT
Solve each problem.
1. Michelle and Corazon are
comparing their heights. If
Michelle’s height is 120
3
4
cm. and
1
3
Corazon’s height is 96
cm. What is
the difference in their heights?
2. Angel bought 6
3
4
meters of silk,
3
1
2
meters of satin and 8
1
2
meters of
velvet. How many meters of cloth
did she buy?

95. Franz Jeremiah G. Ibay

96. There are 2 ways of adding or
subtracting decimals.
1. Express the decimal numbers
in fractions then add or
subtract as described earlier.
2. Arrange the decimal numbers
in a column such that the
decimal points are aligned,
then add or subtract as with
whole numbers.

97. 1. Express the decimal numbers
in fractions then add or
subtract as described earlier.
Example:
Add: 2.3 + 7.21
=2
3
10
+7
21
100
=2
30
100
+7
21
100
= 2+7 +(
30+21
100
)
=9 +
51
100
=9
51
100
=9.51

98. Example:
Subtract: 9.6 – 3.25
=9
6
10
− 3
25
100
=9
60
100
− 3
25
100
= 9−3 +(
60 − 25
100
)
=6 +
35
100
=9
35
100
= 6.35

99. 2. Arrange the decimal numbers in a
column such that the decimal
points are aligned, then add or
subtract as with whole numbers.
Example:
Add: 2.3 + 7.21 Subtract: 9.6 – 3.25
2.3 9.6
+7.21 - 3.25
9.51 6.35

100. Perform the indicated operation.
1. 1,902 + 21.36 + 8.7
2. 45.08 + 9.2 + 30.545
3. 900 + 676.34 + 78.003
4. 0.77 + 0.9768 + 0.05301
5. 5.44 – 4.97
6. 700 – 678.891
7. 7.3 – 5.182
8. 51.005 – 21.4591
9. (2.45 + 7.89) – 4.56
10. (10 – 5.891) + 7.99

101. ASSIGNMENT
Solve each problem.
1. Helen had P7500 for shopping
money. When she got home, she
had P132.75 in her pocket. How
much did she spend for shopping?
2. Ken contributed P69.25, while John
and Hanna gave P56.25 each for
their gift to Teacher Daisy. How
much were they able to gather
altogether?

102. 3. Ryan said, “I’m thinking of a number
N. If I subtract 10.34 from N, the
difference is 1.34.” What was
Ryan’s number?
4. Agnes said, “I’m thinking of a
number N. If I increase my number
by 56.2, the sum is 14.62.” What
was Agnes number?
5. Kim ran the 100-meter race in
135.46 seconds. Tyron ran faster by
15.7 seconds. What was Tyron’s
time for the 100-meter dash?

103. Franz Jeremiah G. Ibay

104. MULTIPLICATION OF
RATIONAL NUMBERS IN
FRACTION FORM
• To multiply rational numbers in
fraction form, simply multiply
the numerators and multiply the
denominators.
a
b
∙
c
d
=
ac
bd
, where b ≠ 0 and d ≠ 0

105. DIVISION OF RATIONAL
NUMBERS IN FRACTION
FORM
• To divide rational numbers in
fraction form, take the
reciprocal of the divisor(second
fraction) and multiply it by the
first fraction.
a
c
a
d
ad
÷
=
∙
=
, where b, c
b
d
b
c
bc
and d ≠ 0

106. EXAMPLES
Divide.
a.
8
11
÷
2
3
=
8
11
∙
3
2
=
2 ∙4∙3
11∙2
=
12
11
=1
1
11

107. EXAMPLES
Multiply.
a.
3
7
∙
2
5
=
3 2
7 5
=
6
35

108. Find the product or quotient of the following.
1.
5
6
∙
2
3
=____ 6. 20 ÷
2
3
=____
2. 7∙
2
3
=____ 7.
5
12
÷ −
3
4
=____
3.
4
20
∙
2
5
= ____ 8.
5
50
÷
20
35
=____
4. 10
5
6
∙3
1
3
= ____ 9. 5
3
4
÷6
2
3
=____
5. −
9
20
∙
25
27
= ____ 10.
9
16
÷
3
4
÷
1
6
=____

109. SEATWORK
1. Julie spent 3
1
2
hours doing her
assignment. Ken did his
assignment for 1
2
3
times as
many hours as Julie did. How
many hours did Ken spend
doing his assignment?
2. How many thirds are there in
six-fifths?

110. 3. Hanna donated
2
5
of her
monthly allowance to the Iligan
survivors. If her monthly
allowance is P3500, how much
did she donate?
4. The enrolment for this school
year is 2340. If
1
6
are
sophomores and are seniors,
how many are freshmen and
juniors?

111. MULTIPLICATION OF
RATIONAL NUMBERS IN
DECIMAL FORM
1. Arrange the numbers in a vertical column.
2. Multiply the numbers, as if you are
multiplying whole numbers.
3. Starting from the rightmost end of the
product, move the decimal point to the left
the same number of places as the sum of
the decimal places in the multiplicand and
the multiplier.

112. DIVISION OF RATIONAL
NUMBERS IN DECIMAL
FORM
1. If the divisor is a whole number, divide the
dividend by the divisor applying the rules of
a whole number. The position of the
decimal point is the same as that in the
dividend.
2. If the divisor is not a whole number, make
the divisor a whole number by moving the
decimal point in the divisor to the rightmost
end, making the number seem like a whole
number.

113. ACTIVITY
Perform the indicated operation:
1. 3.5 ÷ 2
2. 3.415 ÷ 2.5
3. 78 x 0.4
4. 3.24 ÷ 0.5
5. 9.6 x 13
6. 27.3 x 2.5
7. 9.7 x 4.1
8. 1.248 ÷ 0.024
9. 53.61 x 1.02
10.1948.324 ÷ 5.96

114. ASSIGNMENT
Answer Mathematics 7: Learner’s
Module, pp. 58, Letter B only.